Stanley decomposition

In commutative algebra, a Stanley decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by Richard Stanley (1982).

Definition

Suppose that a ring R is a quotient of a polynomial ring k[x1,...] over a field by some ideal. A Stanley decomposition of R is a representation of R as a direct sum (of vector spaces)

R = ⨁ α x α k ( X α ) {\displaystyle R=\bigoplus _{\alpha }x_{\alpha }k(X_{\alpha })}

where each xα is a monomial and each Xα is a finite subset of the generators.

See also

  • Rees decomposition
  • Hironaka decomposition

References

  • Stanley, Richard P. (1982), "Linear Diophantine equations and local cohomology", Invent. Math., 68 (2): 175–193, doi:10.1007/bf01394054, MR 0666158
  • Sturmfels, Bernd; White, Neil (1991), "Computing combinatorial decompositions of rings", Combinatorica, 11 (3): 275–293, doi:10.1007/BF01205079, MR 1122013


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